In this article, a relation between a gap $d_{k}$ and divisors of composite numbers between $p_{k}$ and $p_{k+1}$ is established. Source: http://arxiv.org/abs/math/0506196v3

Let $\rho = \sigma + i \tau$ be a nontrivial zero of the Riemann zeta-function. In this note, we first present a certain condition on the zero-free region of the Riemann zeta-function such that if it were shown to be satisfied, then it would follow that $\sigma \to 1/2$ as $\tau \to \infty$. Furthermore, we discuss an argument of computational nature regarding the Riemann hypothesis. Source: http://arxiv.org/abs/0706.0357v44

In the paper, we first prove a sufficient condition for the Riemann hypothesis which involves the order of magnitude of the partial sum of the Liouville function. Then we show a formula which is curiously related to the proved sufficient condition. Source: http://arxiv.org/abs/0906.4155v7

Let $\lfloor t \rfloor$ denote the greatest positive integer less than or equal to a given positive real number $t$ and $\vartheta(t)$ the Chebyshev $\vartheta$-function. In this paper, we prove a certain asymptotic relationship involving $\vartheta(t) - \lfloor t \rfloor $ and $t^{1/2}$. Source: http://arxiv.org/abs/0806.1571v4

Let $p_{k}$ denote the $k$-th prime and $d(p_{k}) = p_{k} - p_{k - 1}$, the difference between consecutive primes. We denote by $N_{\epsilon}(x)$ the number of primes $\leq x$ which satisfy the inequality $d(p_{k}) \leq (\log p_{k})^{2 + \epsilon}$, where $\epsilon > 0$ is arbitrary and fixed, and by $\pi(x)$ the number of primes less than or equal to $x$. In this paper, we first prove a theorem that $\lim_{x \to \infty} N_{\epsilon}(x)/\pi(x) = 1$. A corollary to the proof of the theorem... Source: http://arxiv.org/abs/0809.3458v6